Asymptotics for periodic systems

Lassi Paunonen; David Seifert

doi:10.1016/j.jde.2018.11.028

Home/Archive/arXiv

arXiv · 1704.08081

Asymptotics for periodic systems

Lassi Paunonen·David Seifert

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a so-called Ritt operator under a natural 'resonance' condition. This allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions. In particular, we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential. We illustrate our general results by applying them to concrete problems including the one-dimensional wave equation with periodic damping.

Explore related subjects

math.FA math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Lassi Paunonen, David Seifert. 2018-11-16. Asymptotics for periodic systems. https://doi.org/10.1016/j.jde.2018.11.028

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Vector-valued partial sums on unbounded Vilenkin systems

Let $\Gm=\prod_{k\ge0}\mathbb Z_{m_k}$ be a Vilenkin group that is not necessarily bounded, i.e., $\sup_k m_k=\infty$. We prove that, for every UMD Banach space $X$ and every $1<p<\infty$, the Vilenkin partial-sum operators are uniformly bounded on $L^p(\Gm;X)$, with a bound depending only on $p$ and the UMD constant of $X$, and not on $\mathbf m$. This resolves an open problem arising from the work of Clément et al.~\cite{ClementDePagterSukochevWitvliet2000} and later recorded explicitly in the book of Hytönen et al.~\cite[p.~362]{HNVWI}. The proof reduces the partial-sum estimate, via a Paley conjugation identity and a tangent-sequence decoupling argument, to a decoupling inequality for Fourier projections on finite cyclic groups, which appears to be new. The same approach also yields $\mathcal R$-boundedness for the family of partial-sum operators associated with the finer block decomposition, thereby resolving another related problem communicated to us by Fedor Sukochev.

math.FA↗

arXiv2026

A Complex Geometric Approach to the Discrete Gabor Transform and Localization Operators on the Flat Torus

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann-Fock spaces can be expressed as theta functions (or equivalently line bundles on Abelian varieties). We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get necessary conditions which hold in higher dimensions and can expand the known results in the one dimensional case, the primary tool being the theorem of the square. We also give an application to asymptotics of restriction operators which arises via the asymptotic behavior of Bergman kernels and Toeplitz operators for high tensor powers of line bundles and find that time frequency restriction operators on the flat torus will exhibit "plunge" behaviors similar to those of time frequency restriction operators in other contexts.

math.FA↗

arXiv2026

On a minimal Andô dilation for a pair of strict contractions

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on $\mathcal H \oplus_2 \ell_2(\mathcal H \oplus_2 \mathcal H)$ for a commuting pair of strict contractions on a Hilbert space $\mathcal H$. In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions $(T_1,T_2)$ on a normed space $\mathbb X$ provided that the function $A_{T_i}: \mathbb X \rightarrow \mathbb R$ given by $A_{T_i}(x)=(\|x\|^2-\|T_ix\|^2)^{\frac{1}{2}}$ defines a norm on $\mathbb X$ for $i=1,2$.

math.FA↗